Resonance Raman Scattering of Light from a Diatomic Molecule

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May 1976

Resonance Raman scattering of light from a diatomic molecule

D. L. Rousseau Bell Laboratories, Murray Hill, New Jersey

P. F. Williams University of Nebraska - Lincoln, [email protected]

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Rousseau, D. L. and Williams, P. F., "Resonance Raman scattering of light from a diatomic molecule" (1976). P. F. (Paul Frazer) Williams Publications. 24. https://digitalcommons.unl.edu/elecengwilliams/24

This Article is brought to you for free and open access by the Electrical & Computer Engineering, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in P. F. (Paul Frazer) Williams Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Resonance Raman scattering of light from a diatomic molecule D. L. Rousseau Bell Laboratories. Murray Hill, New Jersey 07974 P. F. Williams Bell Laboratories, Murray Hill, New Jersey 07974 and Department of Physics, University of Puerto Rico, Rio Piedras, Puerto Rico 00931 (Received 13 October 1975)

Resonance Raman scattering from a homonuclear diatomic molecule is considered in detail. For convenience, the scattering may be classified into three excitation frequency regions-off-resonance Raman scattering for inciderit energies well away from resonance with any allowed transitions, discrete resonance Raman scattering for excitation near or in resonance with discrete transitions, and continuum resonance Raman scattering for excitation resonant with continuum transitions, e.g., excitation above a dissociation limit or into a repulsive electronic state. It is shown that the many differences in scattering properties in these three excitation frequency regions may be accounted for by expressions derived from simple perturbation theory. Scattering experiments from molecular iodine are presented which test and verify the general scattering theories. Spectral measurements, time decay measurements, and pressure broadening measurements were made on I, in the discrete resonance Raman scattering region; and spectral measurements at several excitation frequencies were made in the continuum resonance Raman scattering region. In each cace calculations based on general theories correctly describe the experimental data.

I. INTRODUCTION behavior in these three regions is so great that they have been used to classify1 the re-emission into normal Over the past few years a great deal of interest has Raman scattering, resonance fluorescence, and reso- developed in the understanding of resonance Raman nance Raman scattering, respectively. This empirical scattering and its application to problems in chemistry, physics, and biology. Most of these studies have been classification, resulting from the marked differences in the scattering behavior, have led to a popular belief2 devoted to either solids or complex biological mole- that fundamental theoretical differences exist between cules, and relatively little attention has been given to Raman scattering, resonance Raman scattering, and simple gaseous molecules. Resonance Raman scat- resonance fluorescence. As pointed out recently by tering is generally not well understood, and the work on Behringer, 3*4 this problem is not new, but dates back solids and complex molecules has done little to clarify to the turn of the century with the question of the dif- the basic mechanisms. In solids, the lack of success ferences between ordinary Rayleigh scattering and may be attributed to complicated electronic structures, resonance Rayleigh scattering. Since this time, two impurities, anisotropies, and ill defined electron- schools of thought have evolved concerning the origins phonon interactions. Similarly, biological molecules of resonance and nonresonance scattering. The first often have strong complex absorption bands; vibrational asserts that the long-lived resonance fluorescence pro- modes may not be readily assigned; broad fluorescence cess is a different physical process than short-lived frequently obscures the discrete structure; and the resonance scattering. The distinction between spectra have a marked excitation frequency dependence. am an the processes is made by asserting that in resonance In contrast, the electronic structure of some small fluorescence the excitation and re-emission may be molecules are quite simple and very well known. divided into a real absorption followed independently by Therefore, the properties of the light scattering from a real emission. In contrast, it is argued that in Ra- such molecules as the incident laser frequency is man scattering (or Rayleigh scattering) this bisection varied from nonresonance, to resonance with discrete of the elementary processes is impossible. It is con- vibration-rotation levels of an excited state, and then sequently concluded that two distinguishably different at high laser energies to resonance with continuum physical processes account for these phenomena. The states, should serve to elucidate the nature of the other viewpoint is that resonance fluorescence and Ra- fundamental processes. In this paper we discuss the scattering behavior of monochromatic light when the man scattering, including resonance Raman scattering, intermediate states are isolated discrete vibrational- are all one process with the empirical differences in- rotational levels of a diatomic molecule and the scat- terpretable from the excitation frequency differences tering behavior when the intermediate states belong to alone. a continuum. Consideration of the scattering of monochromatic Empirically, very different scattering spectra are light from simple gaseous diatomic molecules clarifies obtained if the incident laser frequency is far from this question in a definitive way. As long as scattering resonance, is in resonance with transitions to discrete amplitudes from nearby states are sufficiently small, states, or is in resonance with transitions to con- compared to the resonant state under consideration, so tinuum states. The differences between the scattering as to be neglected, and as long as quasielastic collision-

The Journal of Chemical Physics, Vol. 64, No. 9, 1 May 1976 Copyright O 1976 American Institute of Physics 3519 D. L. Rousseau and P. F. Williams: Resonance Rarnan scattering

FIG. 1. Classification of Raman scattering according to laser frequency. A. The incident laser frequency is far from resonance with any real electronic transition, so normal Raman scattering is' observed. B. The incident laser frequency is in the region of discrete levels of a single electronic intermediate state. We term this process discrete reso- w nance Raman scattering. C. The incident frequency is in B C the range of a dissociative DISCRETE CONTINUUM continuum. We label this RESONANCE RESONANCE process continuum resonance Raman scattering;.. RAMAN RAMAN RAMAN SCATTERING SCATTERING SCATTERING

a1 broadening effects are small, no distinction between Ranzan scattering. In the center, when the incident resonance fluorescence and resonance Raman scatter- frequency is in the region of discrete vibronic transi- ing may be made. From perturbation theory a general tions we term the re-emission discrete resonance expression may be derived which is valid for scatter- Raman scattering independent of whether or not the ing of monochromatic light independent of the excita- incident frequency is at exact resonance. To the right, tion frequency. The spectrum of scattered light obeys when the incident frequency is above the excited state this simple formalism and varies in a predictable sys- dissociation limit in a continuum region, we label the tematic way with excitation frequency. From the theory re-emission as continuum resonance Raman scattering. it is clear that even for exact resonance the excitation To illustrate the general light scattering principles and the re-emission process are not independent, be- discussed in this paper we have selected molecular cause the emitted photon retains the phase information iodine as an example. I, was chosenfor the experiments of the excitation step. because it is nearly ideally suited both experimentally At exact resonance, the scattering amplitude diverges and theoretically. First, iodine's electronic structure owing to the pole in the denominator, and a damping and the energies of the vibrational-rotational levels are term must be included. If we take the relevant matrix very well known6 from fluorescence, predissociation, elements to be real, then doing so introduces an imag- and photofragmentation experiments. Secondly, io- inary component into the scattering amplitude which is dine is a single isotope, homonuclear diatomic mole- strongly peaked at this resonant frequency. It then be- cule with only a single vibrational degree of freedom. comes tempting to classify the real part as resonance Finally, as illustrated in Fig. 2, the presence of an Raman scattering and the imaginary part as resonance excited electronic state with a dissociation limit at 20, fluorescence. This arbitrary separation into ill-de- 162 cm-I above the bottom of the ground state well, fined processes is confusing, however, since each term makes it possible to investigate resonance processes results from the same perturbation expression. The below the dissociation limit (in the discrete resonance contributions from each term are most clearly seen in Raman scattering region) and above the dissociation scattering from continuum intermediate states, in limit (in the continuum resonance Raman scattering which case the real and the imaginary terms contribute region) with conventional argon and krypton ion laser about equally to the scattering intensity and no physical sources. differences between the scattering resulting from the two contributions may be made. Furthermore, in con- As was indicated in Fig. 1, analysis of light scat- sidering resonance with a single sharp intermediate tering spectra divides naturally into three distinct re- state the scattering characteristics vary in a smooth gions with very different characteristics. The char- continuous manner according to the Lorentzian line acteristics' from each of these regions are listed in shape function of that state, and the question of where Table I and discussed in greater detail below. resonance fluorescence ends and resonance Raman In the case of ordinary nonresonance Raman scat- scattering begins becomes obviously meaningless. tering,-. the reradiation is characterized by the follow- For ease of discussion we wish to classify the range ing: (1) The Rayleigh line is strong and the funda- of laser frequencies into three distinct regions. This mental vibrational transition (Av = 1) is weak. The classification is illustrated in Fig. 1. On the left, higher overtones are weaker yet and are generally not when the incident frequency is far from resonance with observed at all in the spectrum; (2) The scattered band a real transition we label the re-emission as normal envelope is relatively broad, may have some structure,

J. Chem. Phys., Vol. 64, No. 9, 1 May 1976 D. L. Rousseau and P. F. Williams: Resonance Raman scattering

ropy in the polarizability. At higher incident laser frequencies in the region of discrete resonance Raman scattering the properties of the re-emission are significantly different: (1) The overtones may be of comparable intensity and in fact may be as strong as the Rayleigh line. The ratio of the intensity of one overtone to the next varies erratically; (2) With sufficiently narrow excitation each overtone consists of only a few very sharp lines. Their widths are limited by the combination of the natural, collisional, and Doppler broadening in addition to possible nuclear hyperfine splitting. The structure within each overtone changes erratically with small incident frequency shifts; (3) The scattered intensity varies wildly with excitation frequency; (4) The Stokes-anti-Stokes ratio is not simply related to the ground state population factors and in fact the anti-Stokes intensity may be greater than the Stokes intensity in some cases; (5) At exact resonance the scattering time is long and is typically in the range of 10-5-10'8 sec, although as INTERNUCLEAR SEPARATION (i) the incident frequency is moved away from exact res- FIG. 2. The X and the B electronic state potential functions onance the scattering time becomes very short; (6) of molecular iodine. The dashed line at 20 162 cm-' is the dis- Addition of foreign gas strongly quenches the intensi- sociation limit of the B state. The numerical listing on the ty; (7) Generally the scattering of the Q branch is de- left corresponds to the excitation energies from v" = 0 of the polarized, with a depolarization ratio of $. ground state of the prominent argon and krypton laser lines (expressed in angstroms). In the region of continuum resonance Raman scattering the observed spectra are substantially different from those in the discrete resonance case: (1) Again and exhibits no dependence on the excitation frequency; there are numerous overtones with comparable intensi- (3) The scattering intensity varies slowly and smoothly ty to the fundamental and the Rayleigh line, The in- (an w4 dependence) with excitation frequency; (4) The tensity variation from one overtone to the next is now Stokes-anti-Stokes ratio may be calculated from the very regular; (2) The band envelope within each over- Boltzmann factor; (5) The scattering time is very fast; tone region appears broad with considerable structure (6) Addition of foreign gas does not quench the scat- which smoothly changes as the excitation frequency is tering intensity; (7) The Q branch depolarization ratio varied; (3) The scattering intensity varies with the is small, with the exact value determined by the anisot- excitation frequency in a smooth systematic way; (4)

TABLE I. Excitation-frequency dependent characteristics of light scattering from diatomic molecules.

------Nonresonance Discrete resonance Continuum resonance Characteristics Raman scatterine Rarnan scatterinn Raman scatterine 1. Overtone No overtones Strong; irregular Strong; systematic intensity variation variation 2. Band Broad; structured; Sharp; erratic Broad; structured; envelope no frequency frequency dependence systematic frequency dependence dependence 3. Frequency wi Strong and erratic Strong and systematic dependence of intensity

4. Stokes-anti- cc Boltzman factor Not solely population Not solely population Stokes ratio dependent dependent 5. Scattering Short S 10"~-10"~ Variable - 10~~-10-'~ Short S; 10"~sec time sec sec 6. Quenching Unquenched Quenched Unquenched behavior 7. Q-branch Less than $ 2 Less than depolarization ratio

J. Chem. Phys., Vol. 64, No. 9, 1 May 1976 3522 D. L. Rousseau and P. F. Williams: Resonance Raman scattering

The Stokes-anti-Stokes ratio is not solely related to first convenient to separate out the vibrational wave- the ground state population factors but varies sys- function by using the Born-Oppenheimer approximation, tematically with incident frequency; (5) The scattering time is fast; (6) Addition of foreign gas does not quench dJe,v=Qe(r, R)ve,v(R) (3 the intensity; (7) The depolarization ratio is small and Here qe,, is the total vibronic wavefunction; Qe(r,R) is may be calculated from the electronic structure of the the electronic wavefunction (with electronic quantum molecule. number e) which depends on the electron coordinates These disparate properties may be interpreted by r and the nuclear coordinates R. qeVv(R)is a vibrational wavefunction of electronic state e with vibra- using standard second order perturbation theory. In the next section the relevant expressions are derived tional quantum number u. and in the following sections we use these results to In writing Eq. (3) in this form we implicitly ignore discuss the interpretation of the properties listed here. the dependence of the wavefunctions on molecular We also present data obtained on molecular iodine orientation (the angular coordinates of R). Inclusion which tests the validity of several aspects of the for- of this dependence in the matrix elements of Eq. (2) malism. only results in a multiplicative factor which depends on the rotational quantum numbers and on the polariza- II. THEORY tion of the incident and scattered light. This factor is The total scattering intensity I, in photons per mole- responsible for the dipole selection rule on rotational cule per second for a transition from the initial ground transitions, but in most of the following discussion it state, I G), to the final state, IF), may be given by the plays no other role and for clarity of presentation it standard expressi~n~*~ will be ignored. In the discussion of continuum resonance Raman scattering we will want to compare intensities of lines with different rotational quantum numbers, and in Eq. (26) this rotational factor reappears Here c is the velocity of light, I, is the incident inten- as the factor b, . Detailed evaluation of these rotational matrix elements has been carried out most re- sity at frequency u,, and w, is the scattered frequency (~y,,),, is the polarizability tensor for the transition cently by Silverstein and St. peters" and by Jacon and Van Labeke. l1 from I G) to IF) with incident and scattered polariza- tions indicated by a and p, respectively. The second By substituting Eq. (3) into Eq. (2) we may rewrite order perturbation expression for (a,,),, is the matrix elements as follows, (F(P~I)=(~~~,~(R)~~~(^/,~)(61~e,(~, R)'P~,,~(R)) (4) in which expression the electronic matrix element is

M(R) = (eeF(~,R) 10 (eer(~tR)) (5 The electronic matrix elements, such as those shown In this expression w,, is the energy spacing between an here, are expected to be weakly varying functions of excited rovibronic state I I) and the ground state I G); the internuclear separation R, so they may be expanded and p is the electron momentum operator. For the in a rapidly converging Taylor series, subsequent discussion we will neglect the second term in Eq. (2), since near resonance it is small in comparison to the first term owing to its large resonance denominator. In addition, for simplicity we will neglect In this equation M(5,) is the electronic matrix element the polarization subscripts. It must be noted that Eq. evaluated at the equilibrium internuclear separation (2) is completely general and is expected to correctly <,, and A5 is the deviation from this separation. For treat both Raman and fluorescent processes within the simplicity, the expansion was carried out in a dimen- limits of pertrubation theory. However, it is very dif- sionless variable 5 where ficult to make meaningful interpretations or calculations from it because the states IF), I G), and I I) must be described by complete rovibronic wavefunctions which depend on both nuclear and electronic coordinates. p is the reduced mass of the oscillator and w, its har- A variety of approximations and expansions are there- monic frequency. fore made at this point, 218*9 after which the basic physical processes often become obscured. In the discus- Using the Taylor expansion we obtain for the zeroth sion presented here we have adopted a very simplistic and 1st order terms in the perturbation series approach, but one which makes very transparent the frequency dependence of the light scattering. This general approach and other more sophisticated ones may be found in a variety of more lengthy discussions of the theories of Raman and resonance Raman scattering. (8) In order to make Eq. (2) more manageable7 it is Now the states I g) and I f) correspond to vibrational

J. Chern. Phys., Vol. 64, No. 9, 1 May 1976 D. L. Rousseau and P. F. Williams: Resonance Raman scattering 3523 wavefunctions of the ground electronic state and l i ) to (fl A5 Ig). This term may be significant and should be the vibrational wavefunctions of the excited electronic included in any detailed study of off-resonant Raman state. For simplicity, we now are considering only scattering, but its inclusion leaves the predicted quali- a single excited electronic state. It should be pointed tative behavior of the scattering unchanged-the Ray- out that the matrix elements of the type (f l i), etc., leigh line is much more intense than the fundamental, correspond to Franck-Condon overlap amplitudes. To etc. Since we are primarily interested in resonant understand the expected intensities in a light scattering scattering, for which case all approximations based on experiment we must determine the magnitude of the two a slowly varying denominator fail completely anyway, terms in Eq. (8) in various limits. a more complete discussion of these approximations would lead us too far afield and we refer the reader to A. Normal Raman scattering Ref. 8, where a complete discussion is presented. First let us consider the condition7 that the incident B. Discrete resonance Raman scattering laser energy is very much less than the energy w,, needed for a real transition, i. e., w, << w,, . This 1. Frequency spectrum represents off-resonance light scattering as indicated In the limit of the laser frequency coinciding with a in diagram A of Fig. 1. Because of the above inequality real transition (w, = w,,) for some l i ) independent of the denominator in both terms may be considered near- whether or not the levels are discrete or continuous, a ly a constant, Q = w,, - w, , and the sum over intermedi- situation completely different from the nonresonance ate states may then be contracted using completeness case arises. In this resonance condition in the sum to give over states in Fq. (8) in no approximation whatsoever are the denominators constant, so the completeness argument fails and all terms may contribute to both the Rayleigh and the Raman scattering. However, because of the rapidly converging Taylor series in which the coefficient of the second term in typically about 3 The vibrational wavefunctions are orthogonal so the orders of magnitude smaller than the first, most of the matrix element in the first term gives a Kronecker Rayleigh and the Raman scattering intensity will now delta function in the inital and final vibrational quantum result from the first term alone i. e. numbers, vi', v;', and if we assume the ground electronic potential is harmonic the second term vanishes unless v;' = v:' i 1. Therefore, for off-resonance light scattering No longer is there a selection rule discriminating against overtones, but instead the overtones are expected to be strong and dependent only on the magnitude of the appropriate Franck-Condon overlap factors. Further simplifications of Eq. (11) may be made for discrete resonance Raman scattering. When the in- where cident frequency is in the discrete resonance region and resonant with a specific initial to intermediate state transition then that transition should dominate the summation and all other transitions are weak in comparison. Equation (11) may then be written = (v:'/2)' l2 if vi' = v;' - 1 . We see from Eq. (10) that the first term results in unshifted scattered radiation (Rayleigh scattering) and the second term gives re-emission shifted from the in- Note that because we are now considering resonance cident photon by plus or minus one vibrational quantum, with a single discrete state, it is necessary to include hence Raman scattering at the fundamental frequency. the damping term, ir, which was omitted from the Because the Taylor series is rapidly converging, the previous formulas for simplicity. The linewidth, Raman term is expected to be very much weaker than r, was of course not important in the off-resonance Ra- the Rayleigh term, and higher order terms in the ex- man case because it was so small in comparison to pansion which result in overtones in the Raman spectrum are expected to be weaker yet. Indeed, from a w,, - w, . From Eq. (1) we then obtain the following expression for the scattering intensity: comparison of the Rayleigh to Raman scattering cross sections in simple gases it appears that the experimental ratio of intensities of the coefficients of the two terms in Eq. (10) is typically about 3 orders of magnitude. From this equation we see that the structurein the The weak variation of the energy denominator in Eq. spectrum of the re-emitted light then will depend on the (8) may be treated by expanding it in a Taylor series specific vibrational and rotational quantum numbers of about some average frequency, Q. Doing so results in the selected excited state. The intensities of the suc- the addition of another term in Eq. (9) proportional to cessive overtones will vary erratically depending on the

J. Chem. Phys., Vol. 64, No. 9, 1 May 1976 3524 D. L. Rousseau and P. F. Williams: Resonance Raman scattering

magnitudes of the Franck-Condon overlap factors con- (a) INCIDENT necting the intermediate and final states. A slight change in incident frequency will bring about resonance with a different discrete state of the complicated excited state vibration-rotational manifold, with different vibrational and rotational quantum numbers so that a completely different appearing re-emission spectrum may be observed. The effect of collisional energy redistribution further complicates this spectrum as will be seen later. (b) SCATTERED

2. Re-emission lifetime While the above discussion described the frequency dependence of the intensity variations for discrete Raman scattering, time dependent excitation intensi- ties12 are needed to understand the re-emission lifetime behavior. Consider a simple three level system such as that shown in Fig. 3. The scattering process FIG. 4. Incident and scattered theoretical pulse shapes. a. consists of the absorption of an excitation photon (fre- Rectangular laser pulse with finite rise and fall times used in the calculations. b. Calculated scattered pulse shapes for quency w,) and the emission of a Raman shifted pho- resonant excitation (Aw=0) and off-resonant excitation ton (frequency w,) with a consequent change in the sys- (Aw>>7). tem state from Ig) to If). Although throughout this section we use Raman scattering as our model in the lifetime calculations, the arguments are equally ap- of the Raman intensity caused by using too narrow a plicable to Rayleigh scattering. We now apply a pulse frequency filter, the spectrometer frequency resolution of light, such as that shown in Fig. 4, of frequency is taken to be sufficiently broad that it passes essential- o, to the system and ask what the time response of ly all of the Raman reradiation. Doing so, we may the Raman shifted re-emission is. For the moment set g(w) = 1. we consider arbitrary pulse shapes, with the only re- striction being that the pulse intensity is zero for t < 0. If the excitation field is given by E,(t) = ei"~'s(t), For weak excitation fields, the time response of the where S(t) = 0 for t 5 0, then the second order transi- Raman shifted radiation may be determined by treating tion amplitude may be written the excitation pulse as a perturbation on the system and using second order time dependent perturbation theory to calculate the probability amplitude for the system to be in state If ), with the emission of a Raman photon where (w,), and the absorption of a laser photon (w,) at time t given that at t = 0 it was in I g). This amplitude will be denoted by (f; w, l q2(t)), where w, is the frequency of the re-emitted Raman photon, and the subscript 2 indicates that the state I $) is correct to second order. Here of, and wig are the energies of the states If) and If the re-emission is observed with a spectrometer l i ), relative to Ig) ; (2I3-l is the natural lifetime of the with frequency resolution function g(w), then the ob- state l i); and the variables of integration, 7, T', cor- served intensity of the Raman shifted radiation has the respond, respectively, to the time of absorption of a following proportionality: laser photon and the time of re-emission of a Raman photon. Although the algebra is somewhat tedious, the inte- In order to eliminate distortions of the time response gral in Eq. (15) may be carried out and Eq. (14) evaluated. It is simpler, however, to note13 that the time dependence of the re-emitted light intensity is determined by the time evolution of the population of the intermediate state, i. e.,

where I (i I ql(t)) 1 is the admixture of the intermediate FIQ. 3. Model three level system for considering discrete state I i) into the state Ig) by the perturbation, E,(t), resonance Raman scattering. w~ and w, are the incident and correct to first order. In analogy with the preceding scattered frequencies, respectively. discussion, this quantity is determined by

J. Chem. Phys., Vol. 64, No. 9, 1 May 1976 D. L. Rousseau and P. F. Williams: Resonance Raman scattering 3525

1 Using the above formulation, other pulse shapes such (i 1 = (i 1 P G(t) , (17) as a cusp function, may be handled with equal facility, We have chosen the shape shown in Fig. 4 because it where does not require the excitation to have been turned on for an infinite time in the past, and since by choosing G(t) = e-rt e-iwit J~~ei(wi-fr)T e -iwLT s(T)~T . T sufficiently large, the deconvolution of the input pulse from the output is relatively easy. Using Eq. (17) in Eq. (16), we obtain for the observed Raman intensity Specifically, the function S(t) is defined by

Upon specification of the pulse envelope function, = (1 - e-7T) e-7(t-T) T5t , (19) S(t), the evaluation of G(t) proceeds simply. For clarity of discussion, we take pulse to be as shown in where T is the pulsewidth, and y-I gives the rise and Fig. 4. Finite exponential rise and fall times are fall time of the electric field of the pulse. Letting Aw assumed in order to minimize the effects of transients. = w,, - w, and 8 = y - r, the general result is

A (d cosAwt - e'r(t-T) cosAw(t - T) - -r (e-rt sinAwt - e'r't-T'sinAw(t - T))P t2T. (20) Putting Eq. (20) into Eq. (18), the time dependence of the re-emission is determined.

To obtain a qualitative understanding of the be- From Fqs. (21) and (22) we find that on resonance havior of I G(t) 1 ', it is desirable to take the excitation the time response is "slow, " being limited by the natural pulse rise and fall times to be much faster than the lifetime of the excited state. Off resonance, the time natural lifetime of the excited state, that is, y>> I'. response is fast, being limited by the temporal line Under this condition for exact resonance, Aw = wit - w, shape of the probing pulse. The question of what the = 0, we get scattering time is in the intermediate region is an in- teresting one. Intuitively, we would expect the lifetime to be governed by the uncertainty principle. In this argument the lifetime At for the re-emission is expected tobe limited by the frequency difference Aw between the excited state transition frequency and the incident frequency, i. e., At = l/Aw. That is, if energy 2 I G(t) I for this case, as shown in Fig. 4, has a rise and is not conserved in the transition to the excited state, fall time characteristic of the natural response time of by an amount Aw, then the time the molecule can spend the system (2r)-'. However, far off resonance, the re- in the state if limited to l/Aw. Unfortunately, efforts sponse is "fast" in that it follows the excitation pulse, to measure such a scattering time are complicated be- as may also be seen from Eq. (20) by taking the case in cause the precise definition of the scattering time be- which the shift from resonance is much greater than comes unclear owing to limitations imposed by the fre- the Fourier broadening of the pulse, i. e. , Aw >> y. quency-time uncertainty principle. Then, In order to measure a scattering time T in an off- resonance experiment, we must sharpen the rise and fall times (2y)-' of our excitation pulse so that TL (2y)-'. If we expect a scattering time T of order l/Aw, then this requires that Aw < y. In this region I G(t)1 as given by Fq. (20) is rather complicated, with a strong The behavior of I G(t) 1 under this condition is also oscillatory component at frequency Aw. The leading shown in Fig. 4. The case for Aw = y is more com- and trailing edges of the response pulse areby no means plicated because the behavior is, in this case, deter- exponential, so that an unambiguous measurement of the mined by the rapidly oscillating sin and cos terms in scattering time is impossible. The cause of difficulty Eq. (20). is easy to uncover. By temporally sharpening our ex-

J. Chern. Phys., Vol. 64, No. 9, 1 May 1976 3526 D. L. Rousseau and P. F. Williams: Resonance Raman scattering citation pulse, we have broadened its frequency spec- later, this prediction is completely consistent with all trum by an amount y2 Aw. We now see re-emission of our experimental results, from a range of excitation frequency components, some of which are resonant with the initial transition. The 3. Collision effects oscillatory behavior results from an interference of The frequency behavior of the scattering of mono- each of these re-emission components. Because even chromatic light in situations where collisions are im- in physically allowable "gedanken" experiments a di- portant was derived first by ~uber'~and later by Mollow rect measurement of the scattering time fails, l4 the and Omont et nl., l6and the theory for the temporal be- question of the detailed nature of its variation with in- havior under such collisions was derived by Huber. '' cident frequency must be approached with some care, In the derivation of the expected spectral re-emission We can say that on resonance (AIJ = O), the scattering properties that take place when molecular collisions time 7 for the Raman process is equal to the natural may occur, l5 the collisional processes were treated lifetime of the excited intermediate state involved, in the impact approximation. We obtain for the Raman and off resonance, TSl/Aw. As we will discuss scattering

We have neglected Doppler broadening and broadening second term in Eq. (23) is redistributed over the full of the ground and final states. The intermediate state width of the pressure broadened excited state, and in- linewidth consists of three Lorentzian contributions: dependent of the excitation frequency, it is centered at yN corresponds to the natural lifetime; y, is the the resonant energy and has a lifetime of l/(yN+y,). broadening resulting from a shorter lifetime caused Obviously this may occur only if the line broadening by inelastic collisions, i. e., those collisions in which mechanism includes a relaxation process which can the molecule is knocked into a different quantum state; take up or remove energy. In gases this occurs through and y, is the contribution from quasielastic collisions, quasielastic collisional broadening where during a col- i. e., those collisions in which the phase of the oscil- lision the phase of the oscillator is interrupted but the lator is interrupted but it remains in the same quantum molecule remains in the same quantum state. Because state. Note that the first term in braces in this equa- this term results in a redistribution of the incident en- tion is equivalent to our previously derived Eq. (13). ergy, to maintain generality we shall refer to it as the redistribution term, We emphasize that the redistri- In the limit of a very low pressure gas y, and y, are bution scattering requires the presence of some excited zero, so the second term in the braces in Eq. (23) state relaxation process. In the absence of such a pro- vanishes. In this low pressure limit, and where only cess only the Raman scattering remains, even for ex- one intermediate state is contributing to the scattering actly resonant excitation. Simple energy conservation amplitude, the first term in Eq. (23) fully characterizes arguments then require the re-emission linewidth to the light scattering. There is a phase memory be- be independent of the excited state linewidth. A similar tween the incident and scattered radiation and for an in- derivation using density matrix formalism was carried dividual molecule the scattering process is coherent regardless of whether or not the incident frequency out by hen'' for solids and in Ref. 19, where the as- coincides with the center frequency of the transition. sumption was made that the ground state linewidths When the incident frequency is moved away from reso- were very narrow, and an expression equivalent to Eq. nance, the intensity of the scattered light varies in a (23) was obtained. Lorentzian manner as defined by Eq. (23) and may be The temporal response of this term under pulsed appropriately termed Raman scattering (resonance excitation was investigated theoretically by ~uber" Raman scattering for exact resonance). and a general expression was derived describing the As the gas pressure is increased, the second term time dependence of the intensity. Evaluating his ex- in the braces in Eq. (23) becomes important. This pression (Eq. 3.11 from Ref. 17) for slightly off- term represents a physically distinguishable process, resonance excitation, we obtain a time decay in the resulting from a disturbance in the excited state such limit of negligible collisional broadening effects of that the phase is randomized. The re-emission in- e-Yt consistent with our previous result. This result tensity for this incoherent process compared to the was obtained by assuming negligible collisional broaden- Raman process is given by the probability for elastic ing and that the natural lifetime r is significantly collisional broadening divided by the probability for longer than the decay time of the excitation pulse y, radiative re-emission without suffering an elastic col- If, on the other hand, the collision effects are large, lision, i. e., y, /(yN+ y,). Since all phase information we obtain a time decay of e-rt. We see then that the is lost in the collision, in contrast to the Raman term contribution to the time decay of those molecules un- this term corresponds to independent absorption and affected by collisions is one that follows the laser emission processes. The emission resulting from the pulse, while the contribution from those that have un-

J. Chem. Phys., Vol. 64, No. 9, 1 May 1976 D. L. Rousseau and P. F. Williams: Resonance Raman scattering 3527 dergone pseudoelastic collisions has the time response expected to be very weak in comparison to the funda- of the intermediate state. mentals. In addition, the spectrum of the fundamentals should be highly structured owing to all the allowed C. Continuum resonance Raman scattering Raman transitions from all the populated vibrational- For excitation into the continuum, 20 just as in the rotational levels. In contrast, in the continuum re- case of discrete resonance Raman scattering, we must gion the scattering according to Eq. (11) should con- consider Eq. (11): sist of numerous overtones of similar intensities and each having a great deal of structure. The scattering intensity in this case is governed by the weighted sums and again we would expect overtones to be strong and of the Franck-Condon overlap amplutides. When the depend only on the magnitude of the appropriate incident frequency lies in the discrete region, the con- Franck-Condon factors. Here, because we are dealing tinuum contribution is now obscured by the strong with a continuum, we may assume the limit of infinitely resonances with specific intermediate states. Again narrow states and it is not necessary to include the there should be numerous overtones but the intensities damping term, However, as will be discussed later, of the various wertones should vary erratically be- it is necessary to somehow accommodate the pole which cause of the variations in the (f I i) overlap factor in occurs for those continuum states at exact resonance. Eq. (12). For excitation into a continuum the resonant denomi- When the incident frequency is varied in normal Ra- nator no longer picks out a single vibrational-rotation- man scattering, because it is far from resonance and a1 state transition. Instead, all populated vibrational- the resonance denominator may be considered to be rotational levels of the ground electronic state are at constant [see Eq. (9)], all of the frequency dependence resonance with some continuum state. The re-emission is left in the wb factor of Eq. (1). On the other hand, observed in a given overtone now, rather than result- for discrete resonance Raman scattering a small ing from a single vibrational-rotational transition, is change in incident frequency causes the resonance to composed of a summation of lines resulting from Ra- occur with a different intermediate state [~q.(12)] man transitions from all the populated levels. This so a completely different spectrum is observed. In large number of superimposed bands has an averaging continuum resonance Raman scattering the spectrum effect on the overtone intensities and frequency de- also changes with excitation frequency, but owing to pendence. An additional difference exists between the the averaging effects discussed the changes are smooth discrete and continuum resonance cases, In the dis- and continuous. The Stokes-anti-Stokes ratio for non- crete case a given line in the re-emission spectrum resonance scattering is proportional to the population results from single excited state energy level. Be- of the state involved. In both discrete and continuous cause of the AJ=* 1 dipole transition selection rule, resonance Raman scattering this ratio is not solely the next closest intermediate state which could con- population dependent because the overlap factors also tribute to the intensity of that line is one separated by affect the experimental ratio, the excited state vibrational spacing. Because of the An upper limit for the scattering time of a Raman resonance denominator this "off-resonance" scattering process is approximately (Aw2+r2)-1/2, where r is the does not contribute significantly to the intensity of the natural linewidth. In the discrete region for exactly line in question. In contrast, though, in the continuum resonant excitation, Aw = 0, so the scattering time is region the spacing of vibrational levels is effectively given by the natural lifetime of the state involved, infinitesimal, and a given re-emission line gets in- typically 10-~-10-' sec. For incident frequencies off tensity not only from the continuum state it is in exact resonance with the discrete transition and for normal resonance with but also from other nearby continuum off-resonance Raman scattering, the upper limit of the states that are only slightly off resonance. This serves scattering time is given by ~/AW (typically 10-14-1~-15 as an additional averaging effect on the intensity. In sec in the normal off-resonance Raman scattering the continuum resonance case the various overtones, case); and in continuum resonance Raman scattering rather than varying in a haphazard manner as in the an upper limit may be given by the time needed for the discrete case, vary in a systematic manner because atoms to fly apart (- lo-'' sec). Although the definition of those two averaging effects, Similarly, when the of the scattering lifetime becomes unclear in these excitation frequency is changed the form of the spec- situations because a direct measurement by time de- tra change in a very continuous and systematic manner cay can no longer be made, the manifestations of the as certain vibrational-rotational states become weighted shortened time for the molecule-photon interaction do more or less heavily by the combination of the Franck- have a predictable effect on other physically measur- Condon factors and the resonance denominator. able properties such as foreign gas quenching; namely, D. Conclusions from theory quenching is observed only when the gas collision rate The behavior of the scattering as the incident laser is comparable to the time for re-emission. Conse- frequency is varied from nonresonance, to resonance quently, for off-resonance Raman scattering and for with discrete states, then at higher energies to reso- continuum resonance Raman scattering where the scat- nance with continuum states should now be apparent tering time is fast, quenching of the Raman intensity is and the differences in the properties listed in Table I observed only at very high foreign gas pressures. On reconcilable. In particular, according to Eq. (10) for the other hand, for discrete resonance Raman scat- nonresonance Raman scattering wertones would be tering at exact resonance quenching is observed at

J. Chern. Phys., Vol. 64, No. 9, 1 May 1976 3528 D. L. Rousseau and P. F. Williams: Resonance Raman scattering very low foreign gas pressures (fractions of an atmo- and the light was focused into the gaseous reagent sphere). As the incident frequency is moved away grade I, with a short focal length lens. The scattered from exact resonance in the discrete region, the time light was gathered at right angles and focused onto the for re-emission becomes faster and quenching is only slits of a Spex 1401 Double Monochromator. The dis- observed at higher pressures (- 1 atm). persed light was detected with an ITT FW-130 photomultiplier, amplified, and displayed on a strip chart The depolarization ratio of Sand 0 rotational branches recorder. is independent of the incident frequency, but the depolarization factor of the Q branch varies with the ex- A. Discrete resonance Raman scattering citation frequency. For normal Raman scattering it is typically 0.001-0.01 and depends of the anisotropy of In these experiments the reagent grade I, was dis- the polarizability. 22 In resonance the depolarization tilled several times prior to distillation into a sealed behavior has been treated recently by silverstein" and quartz spectroscopic cell. The pressure of the I, in by Jacon and Van Labeke. " In each case they used this cell was controlled by placing a low temperature perturbation theory and showed that, for high J, the bath around a side arm on the cell. Typically, how- discrete resonance Raman Q branch depolarization ever, experiments were run with the side arm either ratio is $ and for continuum resonance Raman scatter- at room temperature giving an I, pressure of - 0.25 ing it is $ if the angular momentum along the inter- Torr or at ice temperature giving a pressure of - 0.03 nuclear axis is zero. Torr. In these experiments great care was taken to insure that no residual air remained in the sample cell, Ill. EXPERIMENTAL RESULTS AND DISCUSSION as it would yield spurious data due to quenching. Spectral and lifetime measurements were made on molecular iodine to test the scattering theories, The 1. Frequency spectrum well-established electronic states of I, make it a very The re-emission from I, in the region of discrete suitable choice for such studies. As shown in Fig, 2, transitions is characterized by a series of overtones the excited state potential function B(~II,+,) has a dis- of erratically varying intensity as shown in Fig, 5. sociation limit 20 162 cm-' above the X('C ,+,) ground This spectrum was obtained by single mode excitation state. As such, there are several argon ion laser in the 5145 A region. The vibrational fundamental lines with frequencies above the dissociation limit and appears at about 213 cm" and each overtone is sepa- several argon ion and krypton ion laser lines below the rated by about this frequency. We have arbitrarily limit, making the experimental studies in both re- stopped the spectrum after the Av'' = 9 line, but sev- gions straightforward, Numerous scatteringatz4(or eral more overtones may be readily seen, Each of the fluorescence) studies have been carried out in both overtones results from a transition to a specific in- regions but in general the emphasis has not been on termediate excited u' state, in this case u' = 58 of the detailed frequency dependence studies. We have, B state, and then a re-emission to the ground state under high resolution conditions, studied the spectral with the overall change in the vibrational quantum num- changes occurring with several different excitation ber indicated in the figure. When the rotational selec- frequencies above the dissociation limitz0; and we have tion rule for a dipole transition is AJ= * 1, as it is in measured the spectral and temporal changesz5as the I,, a pattern of doublets is observed in the emission, incident laser frequency was tuned away from resonance as seen here. This occurs because after the initial with a transition to a single discrete state. Our re- transition to the excited state with rotational quantum sults are consistent with the preceding theoretical dis- number J' re-emission to final states with J" = J'+ 1 cussion and with the characteristics listed in the sec- or J" = J' - 1 are possible, resulting in a doublet. The ond and third columns of Table I. overall change in the ground state vibrational quantum The spectral measurements were obtained in a stan- number J" then will be 0 and + 2 if the initial state is dard manner. Coherent Radiation Models 52 and 53 ar- J' - 1 (designated an R transition) and it will be 0 and gon ion lasers were used as the excitation sources, - 2 if the initial state is J'+ 1 (designated a P transi-

FIG. 5. Survey spectrum of discrete resonance Raman scattering from !2. The laser was set at 5145 A and single AV-1 moded with an etalon.

I,_500 FREQUENCY ( cm-I )

J. Chern. Phys., Vol. 64, No. 9, 1 May 1976 D. L. Rousseau and P. F. Williams: Resonance Raman scattering 3529

entirely and a triplet is now observed. The triplet is seen because the laser is now at resonance with two transitions-one a ~[43'-0" ~(13)]and one an R[43'- 0" ~(15)]resulting in overlapping Q branch AJ"= 0 and 0 branch (AJ"= - 2) and Q branch and S(AJ" = + 2) branch contributions, respectively.

2. Re-emission lifetime The lifetime measurements were made on an appara- tus built and describedzs by R. Z. Bachrach in which the time decay is measured by a delayed coincidence technique. Again, Coherent Radiation Model 52 and 53 argon ion lasers were used as the excitationsources. The laser beam was modulated acoustooptically to give rectangular pulses with a 3 nsec rise time although the time response of the electronics used in these experiments was not that fast. The width of the pulse could be varied continuously from about 20 nsec to several psecs, but most measurements were made with 100 nsec pulses. The scattered light, after dispersion by the Spex monochromator, was detected by the FW-130 photomultiplier. After feeding the signal to a time-to- amplitude converter, a decay curve was then obtained on a pulse height analyzer. Lifetimes ranging from about a 10 nsec lower limit could readily be measured. FREQUENCY (cm-') All the discrete resonance experiments reported here were carried out in the 5145 A region, where several FIG. 6. Re-emission spectra of the vibrational fundamental I, absorption lines may be brought into resonance by region of I2 excited ino the region of discrete resonance Raman tuning the single mode laser through the Doppler pro- scattering with 5145 A. The laser mode spacing is about 130 MHz. The top spectrum, labeled Mode t 20, was taken at the file of the laser line. The laser tunes with discrete same laser frequency as the spectrum in Fig. 5. mode jumps of -95 MHz for the Model 53 laser, of - 130 MHz for the Model 52 laser. A total frequency range of nearly 10 GHz is available from this mode selection. In the lifetime measurements the spectrom- tion). The specific re-emission seen in Fig, 5 results eter resolution was maintained at 2 cm-' or better so from an initial transition from the u" = 1 vibrational that a single re-emission line could be isolated. All and J" = 99 rotational level of the X state to the u' = 58, data were obtained in the frequency region of the vibra- J' = 100 level of the B state and is therefore termed tional fundamental (a frequency shift from the incident a 58' - 1" R(99) transition. laser of about 213 cm-I). As discussed in conjunction with Eq. (12), for dis- To study the behavior of the lifetime on and off reso- crete resonance Raman scattering only one inter- nance the spectrometer was set to 216 cm'l to isolate mediate state dominates the scattering, resulting in the - the S branch of the triplet shown in the bottom of Fig. re-emission doublets seen here. The intensity varia- 6. The modes of the incident laser were then changed tion between overtones results from variation in the so as to at first be on resonance with the R(15) transi- Franck-Condon overlap factors between this intermedi- tion resulting in this re-emission and then to gradually ate state and the final state, i. e., (f l i). The addi- move out of resonance with the transition. Although the tional structure seen near some of the overtones results laser was being shifted away from resonance, because from collisional transfer lines and re-emission lines the spectrometer was set at a specific frequency the from additional intermediate states that are only slight- behavior of a single discrete intermediate state was ly out of resonance. In the spectra reported here the still being studied without interference from other linewidth of the individual lines is limited by the spectrom- transitions. Similar experiments at -213 cm-I were eter resolution, but in line shape measurements with done on the Q branch [resulting from both the R(15) and a Fabry-Perot spectrometer the expected Doppler plus P(13) transitions]. The time decay results25of the S hyperfine widths have been observed. branch are shown in Fig. 7. The laser pulse in these By changing laser modes it is possible to change measurements was 100 nsec wide. The 0.0 GHz spec- from resonance with one intermediate state transition trum was obtained by adjusting the laser frequency to to resonance with another. In Fig. 6 a higher resolu- obtain the maximum intensity of the re-emission from tion spectrum of the vibration fundamental region of the P(13) and R(15) transitions, i. e., to be precisely Fig. 5 is shown (labeled Mode # 20) and compared to a at resonance. The other time decay measurements spectrum obtained with a laser mode shifted by - 2.5 were made by shifting the laser modes away from this GHz (labeled Mode # 0). In shifting the frequency by resonance position. The l/e lifetime (- 300 nsec) of only 2.5 GHz, the emission doublet has disappeared the on resonance (0.0 GHz in Fig. 7) is significantly

J. Chern. Phys., Vol. 64, No. 9, 1 May 1976 3530 D. L. Rousseau and P. F. Williams: Resonance Raman scattering

Further off resonance at 2.2 GHz in Fig. 7, the re- 1-----r-7 --r---- emission is dominated by the fast response and only a LASER PULSE 1 !": 1.2 GHz weak residual slow response remains. Decay curves 1 :. similar to the 2.2 GHz curve were obtained from 1.6 GHz off resonance to 2.6 GHz. Because of the weakness of the re-emission from this R(15) transition at 2.6 GHz off resonance, measurements beyond this point were not made. In measurements between 0.0 and - 1.5 GHz off resonance the excitation frequency was still in the wings of the Doppler tail, so as the incident frequency was moved away from resonance the intensity of the "longw-lived component progressively became weaker with respect to the "shortM-lived one. However, for incident frequencies beyond 1,6 GHz off resonance where the incident frequency was in the TlME ( psec ) Lorentzian wings of the absorption the relative intensities of the two components no longer changed. FIG. 7. Time decay data for pulsed excitation of molecular iodine. The spectrometer was set to observe the re-emission It is worthwhile to note that the on-resonance (0 GHz from the Raman shifted S branch (- 216 cm-' shift) of the in Fig. 7) predicted time response for y>> r given by 43'-0" R(15) transition. The laser pulse is shown in the upper Eq. (21) and depicted in Fig. 4 is not consistent with left. In the lower left is the time decay data obtained for the response actually observed. Specifically, the resonant excitation and on the right are data for off-resonant leading edge of the pulse is predicted to have a rise excitation shifted by 1.2 GHz and 2.2 GHz from exact resonance. The accumulation time was substantially different for quadratic in (I - e-rt), which is very different from the each plot ranging from a few minutes for the 0 Gflz decay plot dependence observed in Fig, 7. This difficulty re- to several hours for the 2.2 GHz decay plot. sults from two effects. First, in the decay spectrum of Fig. 7, the peak power of the laser was - 50 mW, causing the resonant transition to saturate. In Fig. 8 shorter than that obtained in prior measurements. 29 we present on-resonance time decay data obtained This results from diffusion of iodine molecules out of with - 0.1 mW of power and an I, pressure of 0.25 the narrow illuminated region focused on the spectrom- Torr. Note in particular the difference in the leading eter entrance slit, thereby selectively discriminating against the long-lived events. With larger slits, we recorded lifetimes of - 1 psec, consistent with prior measurements. At 1.2 GHz off resonance there are clearly two different lifetime contributions-one short-lived re-emission, decaying in a time limited by our 10 nsec temporal resolution, and one long-lived component with the same decay constant as the 0.0 GHz measurement. Naively, on the basis of a single physical process one may have expected to see a continuous change in life- LINEAR time in this region and therefore may be tempted to interpret these results at 1.2 GHz as supportive evi- dence for two independent physical processes (long- lived resonance fluorescence and short-lived Raman scattering). However, consideration of the magnitudes of the quantities involved shows that it results from the complicated line shape of the iodine absorption. It must be remembered that a frequency shift as small as 20 MHz from resonance results in an uncertainty- limited lifetime of 10 nsec, the lower limit of our temporal resolution. In addition, the absorption lineshape of iodine is not determined by its natural (radiative and nonradiative) lifetime, but primarily by the Doppler broadening and the hyperfine splitting, resulting in a I 1 I I I I I I I 0 0.5 total linewidth of over 1 GHz. Consequently, incident frequencies in the wings of the absorption tail (see 1.2 TlME (psec) GHz spectrum in Fig. 7) are on resonance (Aw< 100 FIG. 8. Linear and logarithmic display of time response re- kHz) with some components within the Doppler profile, sulting from excitation of room temperature I2 at exact reso- and very far off resonance (Aw>20 MHz) with others, nance. To obtain these data the laser power was lowered to giving the long- (natural lifetime) and short- (< 10 nsec) 0.1 mW to reduce saturation effects. The dashed line corre- lived contributions, respectively. sponds to the position of the excitation pulse.

J. Chern. Phys., Vol. 64, No. 9, 1 May 1976 D. L. Rousseau and P. F. Williams: Resonance Raman scattering 3531 edge of the pulse. In this figure we believe the devia- larger than a natural linewidth for these transitions, tion from linearity in the decay portion of the log plot so that for excitation frequencies near the center of the results from photomultiplier noise. A second effect Doppler broadened profile, the re-emission intensity causing the pulse to deviate from the simple (1 - e-rf)2 from the ensemble is to an excellent approximation dependence predicted by Eq. (21) results from the Dopp- given by !_", I G(t)1 d~w. This integral is easily car- ler broadening. This broadening is typically much ried out using Eq. (20), with the following result:

- (1 e-"') 2y(l (y- 2I')(1 e-")' 1 ~(f)I2daw = 71 - - - - - 1

In Fig. 9 this function is plotted, and in the insert in not be expected until the collision rate equals the re- the upper right it is compared with the Fig. 8 data for emission lifetime. In Fig. 10 we present quenching the first 100 nsec. In the calculation (2r)" was taken data at several different excitation frequencies obtained to be 125 nsec, the measured value in Fig. 8, and (2y)-1 by adding foreign gas, Nz, to the iodine cell. In Fig. was taken to be 3 nsec, The agreement between the 10 the labels on each curve correspond to the frequency observed and calculated spectra is quite satisfactory, shift in laser modes (one mode - 130 MHz) with the P(13) and R(15) transitions. The spectrometer was set In the direct lifetime measurements reported here, to observe the fundamental Raman re-emission at a we conclude that on resonance the lifetime is slow and frequency shift of about 216 cm". As expected, when off resonance it is fast. As discussed, the uncertainty the incident frequency is near the center of the reso- principle allows direct measurement only of an upper nance the foreign gas very rapidly quenches the fluo- limit to the off-resonance lifetime. However, we might rescence. Off resonance considerable foreign gas had expect quenching behavior to serve as an indirect mea- to be added to the cell before a change in the fluores- sure of intermediate lifetimes. By adding a foreign gas cence intensity was seen. One would like to conclude to the I,, quenching of the re-emission intensity would from this that a lifetime for re-emission could be calculated from that pressure, However, that would only be possible if all of the broadening resulted from inelastic collisions, i. e., those lifetime broadening col-

I I I I 0 I 10 100 loo0 FIG. 9. Linear and logarithmic display of time response pre- PRESSURE (mm Hg) dicted from Eq. (241 with a 100 nsec pusle. (23.1-' was taken to be 3 nsec for the laser pulse rise time and (2r)-' was taken FIG. 10. Quenching behavior of the I2 re-emission upon the to 125 nsec for the room temperature I2 response time. The addition of N2 gas. The curves are labeled by laser modes insert is a linear display of the first 100 nsec of the response, shifted from exact resonance with the P(13) and R(15) lines. and for comparison data points taken from Fig. 8 are shown. The laser mode spacing is 130 MHz.

J. Chern. Phys., Vol. 64, No. 9, 1 May 1976 3532 D. L. Rousseau and P. F. Williams: Resonance Raman scattering

the spectral re-emission, high resolution spectra were obtained. The measurements were carried out by di- recting the scattered light througti an interference filter centered at the Stokes frequency shift (- 213 cm-') and into a piezoelectrically scanned Fabry-Perot interferometer with a 10 GHz free spectral range. The mea-

=*- .. .r.. . . . surements were made in a near forward scattering .---:c;?-- -.. ...,:- .*, L ..-.,.-&,.-*. IIIIIII "IIII,, geometry to reduce the effects of Doppler broadening 0 0.5 0 0.5 TIME Gsec) and, as in the lifetime measurements, the excitation frequency was shifted 1.7 GHz away from the center of FIG. 11. Time decay data for the I2 re-emission obtained at the resonant transitions. The resulting spectrum for two different I2 pressures. 0.25 Torr of I, is depicted by the points in Fig. 12. The sharp features in the spectrum result from Raman scattering from the S, Q, and 0 branches (in different lisions in which the molecule is knocked into a com- orders of the interferometer) and the laser line which pletely different quantum state. Pseudoelastic col- leaked through the narrow band filter. In addition to lisions, though, which interrupt the phase but leave the the sharp features there is a broader underlying struc- molecule in the same quantum state, may also take ture. We attribute this to redistribution scattering. place. Broadening from such collisions in an off- The sharp Raman features have widths limited by resonance experiment effectively increases the absorp- Fabry-Perot resolution. In contrast, the redistribution linearly with pressure, but the quenching-decreases tion scattering width regains the full width of the Dopp- the fluorescence linearly with pressure, giving a pres- ler broadened transition. sure independent intensity in the low pressure region. Finally a point is reached where the absorption saturates The solid curve in Fig. 12 is a calculation of the so the overall fluorescent intensity starts to decrease. spectrum obtained in the following manner. The fre- The pseudoelastic collisions then, as originally pointed quencies of the sharp S, Q, and 0 Raman lines were out by St. Peters and Silverstein, 30 give a behavior fitted to coincide with the experimental frequencies and qualitatively the same as one would expect from a life- were assigned the expected 1, 2, 1 relative intensity time broadening mechanism, and conclusions may not ratios. Shifted 1.7 GHz away from each of the sharp be drawn about the system lifetime. Raman lines a redistribution band is located. Its shape

3. Coi/ision effects

To study the origin of the long-lived remnant shown 1 in Fig. 7 at 2.2 GHz, we have made a determination of LASER the effect increased pressure has on the lifetime re- -1 GHZ sponse. The decay curves obtained in Fig. 7 were '' Q obtained at -0.03 Torr of I,. In Fig. 11 the decay 0' curves obtained at - 1.7 GHz off resonance with I, 1\ i pressures of 0.25 Torr and 0.03 Torr are presented. 4 At 0.25 Torr of I, there are two prominent components to the scattering. One has a fast rise and decay (it I follows the excitation pulse) and therefore results from the Raman contribution to the scattering [first term in S # ,i1 braces in Eq. (23)]. The other contribution has an ex- . ponential rise and decay and we attribute it to the redistribution term. It has a lifetime of about 125 nsec, substantially shorter than the natural lifetime, as expected, since at these pressures the lifetime is given

by l/(yN+ ~r)0 As the I, pressure is lowered, y, decreases, so the relative intensity of the redistribution term to that of the Raman term also decreases. Consequently, as IQI seen in Fig, 11, at 0.03 Torr of I, the long-lived con-

tribution is greatly reduced with respect to the short- FREQUENCY lived component, The lifetime for the slow component is longer than that of the redistribution term at the FIG. 12. High-resolution frequency spectrum of 214 cm-' higher pressure although not as long as expected on the Stokes re-emission of I2 obtained with 1.7 GHz off-resonance basis of the natural lifetime. It exhibits a shorter de- excitation. S', Q', and 0' correspond to the positions of the cay constant than expected owing to the spatial resolu- rotational branches obtained with resonant excitation. The tion imposed by the narrow spectrometer slits used in points are the experimental data. The solid line is a theo- the experiment. retical curve, and the dashed line is the redistribution-scattering contribution. The bars at the bottom indicate the 1.7 As a further study of the effects collisions have on GHz shift between each Raman-redistribution pair.

J. Chem. Phys., Vol. 64. No. 9, 1 May 1976 D. L. Rousseau and P. F. Williams: Resonance Raman scattering 3533 was determined from an optical absorption spectrum we obtained and its intensity corrected for an accidental absorption at the Q branch resonance frequency. The resulting spectrum was then folded into a Lorentzian interferometer response function and a best fit to the data was obtained by adjusting the relative intensity of the total Raman to the total redistribution contributions. The resulting redistribution intensity after convolution with the slit function is depicted by the dashed curve in Fig. 12. For the spectra reported in Fig. 12 we found that the redistribution term had an integrated intensity of about 0.8 that of the Raman contribution. We believe the minor remaining discrepancies between the observed and calculated spectra result primarily from additional low intensity bands resulting from inelastic collisional transfer processes.

The possible contribution from inelastic collisions FIG. 13. Time decay spectrum (data points) of a collisional merits more discussion. Experiments were carried transfer line resulting from inelastic scattering. The width of the laser pulse was 100 nsec and its position is indicated by out to determine if such collisions significantly in- the dashed line. fluence the time decay and frequency data. First the spectral structure of the fundamental vibrational transition region (- 180 cm-'-250 cm-' shifted from the laser ponent in the 0.03 Torr spectrum results from the line) was examined in detail. AS may be seen in Fig. same origin, we cannot rule out the possibility that it 6, where part of this spectrum is shown, transfer results from excitation at the resonance frequency due bands from inelastic collisions make only weak contri- to the spectral distribution of the laser pulse.' Ex- butions to the spectrum. If substantial transfer band periments done at lower pressure should elucidate this intensity from inelastic collisions were present in the point. 1 em-' frequency window of the S branch used to measure the time decay, then near the region of the triplet In the off-resonance pressure sensitive lifetime data strong transfer lines would be observed. Indeed, such of Fig, 11 the contribution from redistribution scatter- lines are observed when several millimeters of foreign ing decays exponentially. If this contribution could be gas are added to the iodine. We conclude from their isolated in an experiment with exactly resonant excita- absence in our experiments that inelastic collisional tion, for a 100 nsec pulse used in this work it should transfer processes do not make a significant contri- have a substantially different behavior, reaching a bution to the lifetime data. maximum somewhat after the termination of the excitation pulse due to the time evolution of the prepared Since the relative intensities of the two contributions state. Because of the limited frequency resolution in (instantaneous and long lived) in Fig. 11 for 0.25 Torr our time decay experiments, we have not been able to pressure are approximately equal and since the broad observe this effect for the pseudoelastic collisional and sharp features in Fig. 12 are also of the same processes. However, we have observed it by studying order, we believe that in the frequency spectrum of the time decay of a transfer line resulting from in- Fig. 12 we are in fact observing the same contributions elastic collisions. As shown in Fig. 13, the time de- as in the time decay spectrum, Fig, 11, namely a con- cay curve3' of a collisional transfer line reaches its tribution from pseudoelastic collisions. However, in maximum after the excitation pulse is turned off. the spectral measurements it was necessary to determine if inelastic transfer lines passed by the narrow B. Continuum resonance Rarnan scattering band filter and the Fabry-Perot interferometer in- fluenced the observed spectrum. Therefore we mea- In experimentsz0above the dissociation limit in the sured the total transmission through the narrow band continuum region the gaseous iodine was in a sample cell held at a temperature of about 420 OK. filter with the low resolution (1 cm-I) spectrometer and By setting the temperature of an I, reservoir to about 360 OK, an compared the integrated intensity of the transfer lines Iz pressure of about 25 Torr was maintained. Spectra to that of the S, Q, 0 triplet. From this measurement we found that the integrated intensity of the transfer were obtained at excitation wavelengths of 4965, 4880, bands from inelastic collisions is a factor of 3 too small 4765, and 4579 with a power level of about 500 mW. One half an atmosphere of gaseous nitrogen was in- to account for the redistribution data of Fig. 12. We troduced to one sample cell in order to measure the therefore conclude that the slow decay in Fig. 11 at iodine temperature during a scattering experiment by 0.25 Torr and the broad features in Fig. 12 result from N,. pseudoelastic collisional redistribution processes. monitoring the rotational distribution of the The addition of the N, had a negligible effect on the struc- In the time decay spectra then, we may conclusively ture and the intensity of the re-emission. The iodine assign the long-lived component at 0.25 Torr as re- temperature due to heating in the laser beam was de- sulting from pseudoelastic collisional transfer pro- termined to be 520-620 OKdepending on the excitation cesses. Although we believe that the long-lived com- frequency.

J. Chem. Phys., Vol. 64, No. 9, 1 May 1976 3534 D. L. Rousseau and P. F. Williams: Resonance Raman scattering

FIG. 14. Survey spectrum of continuum resonance Ra- man scattering from IZ. The laser exciption wavelength was 4880 A.

FREQUENCY (WAVENUMEERS)

When the excitation frequency is brought into the con- were obtained by interpolating between turning points tinuum region many wertones are again observed, as which were empirically determined. Since the turning has been reported previ~usly~'~~and as is shown in Fig. point data only included the banded region of the B 14 for excitation with 4880 A. In contrast to the dis- states, we extrapolated the small internuclear distance crete case, the relative intensity of one overtone to the side of the curve with an a+ b/r12 curve. next varies in a systematic way. The structure within Once the potential curves were obtained, continuum each overtone is real and results from the 0, Q, S wavefunctions for the B state and the bounded wave- branches of the multitude of vibrational and rotational functions for the X state were calculated numerically transitions that are occurring. This structure may be on a computer. The wavefunctions were calculated by better seen under higher resolution conditions as shown solving a radial Schrtklinger equation by an equivalent in Fig. 15 for the overtone corresponding to Av" = 5. finite difference equation. 36 From the resulting wave- As was first pointed out by Kiefer and Bernstein, 32 con- functions, overlap integrals between the bounded vibra- sideration of the Fortrat diagram for each of the rotational states of the X potential and the continuum states tional branches readily explains the origin of the struc- of the B potential were calculated numerically. ture in the spectra. In this figure the sharp features are all S branches and their labeling corresponds to the initial and final vibrational levels, respectively, of the Raman transition. Each line consists of several transitions within the S-branch manifold. Owing to the energy dispersion of the Q and0 transitions as seen in a Fortrat diagram, 32 a well-defined band head is absent, and therefore these branches are not well-defined discrete features in the spectrum. Instead, they lead to the broad underlying band seen in each spectrum. In the region of discrete resonance Raman scattering a change in the laser frequency of 1 GHz resulted in a dramatic change in the scattering spectrum, In Fig. 15, in changing the laser frequency from 4965 to 4579 A, over 1000 cm", the appearance of the spectrum has changed in a very continuous and systematic way. In particular, the relative intensity of the various hot bands has changed from a condition at long wavelengths where low v" hot bands dominate to a condition at shorter wavelengths where higher v" hot bands dominate. This systematic variation in which higher hot bands dominate at higher laser energies is true of all the overtones. Such a smooth variation in structure with incident frequency is expected on the basis of Eq. (11) and occurs because of the continuous variation in magnitude of the overlap factors as a function of excited state energy. To more quantitatively interpret these continuum resonance Raman spectra we have numerically calculated a,f in Eq. (11) for depolarized continuum resonance Raman scattering for several overtones and laser frequencies. In this calculation Eq. (11) was evaluated for transitions occurring via the B continuum FREQUENCY SHIFT (cm-$1 states only. We have neglected to consider the dis- FIG. 15. Structure of the Av" = 5 transition for several laser crete states of the B potential well and the continuum frequencies in the continuum resonance Raman scattering re- states of the In,, potential whose shape, indicated gion. The sharp lines in the spectrum are S-branch transi- schematically in Fig. 16, is not well known. The po- tions and the initial and final vibrational quantum numbers for tential curves for both the B ~tate~~*~'and the X state35 each of these lines are assigned in the upper spectrum.

J. Chem. Phys., Vol. 64, No. 9, 1 May 1976 D. L. Rousseau and P. F. Williams: Resonance Raman scattering 3535

Here P is a constant that includes the vibrational and rotational partition functions, B is the iodine X state rotational constant, b, is the rotational branch intensity factor as given by Placzek and Teller, 39 and the zero of energy was taken to be the bottom of the X potential well. Using this expression the Raman intensity for each vibrational rotational transition was calculated. All the rotational levels with significant population (up to J=400) and hot bands originating from the v" = 0 to vf' = 9 ground electronic state levels were included in the calculation for each overtone. The calculated spectra were finally produced by convoluting the spectral intensity with a two wave number rectangular slit function. In carrying out the calculation we have assumed that M(5) is constant. On the basis of calculation^^^ of the electronic transition moment in I, we would expect this "2 3 4 approximation to be valid since M(5) is nearly a con- INTER-ATOMIC SEPARATION (8) stant over the internuclear separation range of interest. FIG. 16. Potential energy curves of 12. The contributions to the total intensity from the real and the imaginary amplitudes in Eq. (25) are oscillatory with laser frequency, as shown in Fig. 17. Here To evaluate the Raman intensity, overlap factors we have plotted the real (solid curve) and the imaginary were calculated between all the relevant ground state (dashed curve) transition amplitude as a function of the vibrational levels and a series of continuum levels. A frequency above the B dissociation limit for the Ra- small enough net of continuum levels was selected so man transition v" = 4 - 7. Note that these amplitudes that werlap factors for continuum wavefunctions at make approximately equal contributions to the intensity energies not calculated could be obtained by interpola- and oscillate in a mutually compatible way-for inci- tion. The potential curves were the J=0 curves. At dent frequencies in which one contribution is small, the high J the shapes of the potentials change, and they both other contribution tends to pass through a maximum. shift towards larger internuclear separation. 37 How- Similar behavior was observed for the other transitions. ever, the effect of the rotational energy term does not substantially affect the shape of either the bottom of the X potential or the high energy region of the B potential near the inside turning point. Since the dominant contribution to the overlap integrals comes from the region near this turning point, in the calculation reported here we have neglected changes in shape of the potential curves and have accommodated the changes in energies of the potentials as a result of the rotational term, by shifting their relative positions by the appropriate rotational energies. In this manner we have calculated the intensities of Raman transitions originating from the first ten thermally populated vibrational levels of the X state for J values up to J=400. 38 Since only one intermediate electronic state is being considered, Eq. (11) may be rewritten to accommodate the singularity as

where p(w,) is the density of continuum states.

Using a,, we may now write Eq. (1) for each transi- 0 2000 4000 6000 8000 tion to include the rotational and vibrational popula- FREQUENCY (cm'') tion factors and the 0, Q, and S brand intensity factors as follows: FIG. 17. Variation of the real and imaginary parts of the transition amplitude with incident laser frequency for the Ra- man transition v" =4 - 7. The solid curve is the real contribution and the dashed curve is the imaginary contribution.

J. Chem. Phys.. Vol. 64, No. 9, 1 May 1976 3536 D. L. Rousseau and P. F. Williams: Resonance Raman scattering

It should be noted that this oscillatory behavior of the imaginary part of the amplitude has been observed di- r O~SERVLC CALCULATED rectly40 in the discrete-continuum emission spectra in the I, E - B transition, In Fig. 18 we show the comparisons of the calculated and observed spectra for the Av" = 8 overtone and in Fig. 19 for the Au" = 3 overtone. The structure in Fig. 18 is similar to the Az)" = 5 overtone shown previously. However, in the Av" = 3 spectrum in Fig. 19 for the 4965 and 4880 A excitations, there is a clear distinction between the S- and Q-branch envelopes. This distinction has disappeared in the 4579 A spectrum. For high energy excitations these separate S and Q envelopes get washed out by the large number of h~tbands in the spectrum owing to the tendency of higher hot bands to become more prominent with higher laser frequencies. On the other hand, at lower excitation frequencies where only the transitions originating from low v" levels are strong, the S and Q envelopes may be resolved.

The calculated spectra for these two sets of overtones I are shown on the right hand side of Figs. 18 and 19. 660 640 620 600 560 "'660 640 620 600 580 FREQUENCY SHIFT (cm-') Since the 4965 A excitation line is less than 100 cm" above the dissociation limit, a spectrum for this wave- FIG. 19. Experimentally observed and theoretically calcu- length was not calculated because contributions to the lated iodine resonant Raman spectra for the Av" = 3 transition. scattering from the discrete states are expected to be S and Q numbers refer to the initial- and final- vibrational large for this excitation line. The frequencies of the state assignments of the S and Q branches. dominant freatures in the calculated spectra agree with those in the observed spectra within experimental error, The absolute intensity of the calculated spectrum for each laser line was scaled to the experimental intensity. Except for this scale factor, these spectra -- __il were calculated with no adjustable parameters. r--- OBSERVED In the calculated spectra, as in the observed spectra, the trend toward the dominance of higher hot bands at higher laser frequencies may be seen. This results from changes in the magnitude of the Franck-Condon factors with incident laser frequency. In all the calculated spectra there is a general weakness of transitions originating from the lowest vibrational states (i. e., v" = 0, 1) as compared with the observed spectra. We believe this discrepancy results from our approximation that the spectral features are completely governed by the B continuum states. We have neglected contributions to the sum in Eq. (11) from the 'II,, repulsive state and from the discrete levels of the B state. The ratio of the intensity of the second to seventh overtone was measured for each laser excitation frequency, and a calculated ratio was also obtained. For spectra taken at 4579 and 4765 A the calculated and measured intensity ratios agreed within experimental error (- 10%). However, at 4880 A the ratios differed by about 50%. Again we believe this difference at 4880 A results from including only the B continuum states in our intermediate-state calculation. We have FREQUENCY SHIFT (cm-') not measured the relative intensities of a given over- FIG. 18. Experimentally observed and theoretically calculated tone with laser frequency because variations in the iodine resonant Raman spectra for the Av" = 8 transition. S absorption coefficient over the range of interest and numbers refer to the initial- and final vibrational state assign- also the thermal-lens effect complicate the experimen- ments of the S branches. tal measurement.

J. Chern. Phys., Vol. 64, No. 9, 1 May 1976 D. L. Rousseau and P. F. Willkm: Resonance Raman scattering 3537

It is expected that the discrete states may make con- Phys. Lett. 32, 76 (1975); S. Mukamel and J. Jortner, J. tributions to the scattering intensities especially for Chem. Phys. 62, 3609 (1975); H. J. Kimble and L. Mandel, laser wavelengths near the dissociation limit. In ad- Opt. Commun. 14, 167 (1975); H. Metiu, J. Ross, and A. dition, recent photodissociation41and optical-absorp- Nitzan, J. Chem. Phys. 63, 1289 (1975). tionS3experiments have indicated that the Inl,re- 13s. L. McCall, Jr. (private communication). 14s. E. Schwartz, Phys. Rev. A 11, 1121 (1975); J. R. Solin pulsive state has a nonnegligible oscillator strength in and H. Merkelo, Phys. Rev. B 12, 624 (1975). the part of the spectrum we are investigating, and 15~.L. Huber, Phys. Rev. 178, 93 (1969). would therefore also make a contribution to the Raman 16~.R. Mollow, Phys. Rev. A 2, 76 (1970); B. R. Mollow in scattering intensity. However, the position of this re- Third Rochester Cogerenee on Coherence and Quantum Op- pulsive state is not well known. Although the contri- tics, (Plenum, New York, 1973), pp. 525-532; B. R. Mol- bution of the state to the resonant Rarnan intensity low, Phys. Rev. A 8, 1949 (1973); and A. Omont, E. W. appears to be small, a quantitative determination of its Smith, and J. Cooper, Astrophys. J. 115, 185 (1972). "D. L. Huber, Phys. Rev. B 1, 3409 (1970). optical properties in this frequency region should be lay. R. Shen, Phys. Rev. B 9, 622 (1974). possible after including the B discrete states in the 19~.L. Rousseau, G. D. Patterson, and P. F. Williams, summation in Eq. (11). We are currently investigating Phys. Rev. Lett. 34, 1306 i1975). the possibility of using some simplified calculational "P. F. Williams and D. L. Rousseau, Phys. Rev. Lett. 30, procedures42to facilitate this study. 961 (1973). "D. G. Fouche and R. K. Chang, Phys. Rev. Lett. 29, 536 ACKNOWLEDGMENTS (1972). "A. Weber, in The Raman Effect (Marcel Dekker, New York, We thank G. D. Patterson, A. Nitzan, and S. H. 1973), Vol. 2, pp. 543-759. Dworetsky for many helpful discussions. 23~.Kiefer, Appl. Spectrosc. 28, 115 (1974). 24~.B. Kurzel, J. I. Steinfeld, D. A. Hatzenbuhler, and G. E. Leroi, J. Chem. Phys. 55, 4822 (1971). 25~.F. Williams, D. L. Rousseau, and S. H. Dworetsb, 'w. Holzer, W. F. Murphy, and H. J. Bernstein, J. Chem. Phys. Rev. Lett. 32, 196 (1974). Phys. 52, 399 (1970). 26~.D. Patterson, S. H. Dworetsky, and R. S. Hozack, J. 'M. Mingardi and S. Siebrand, J. Chem. Phys. 62, 1074 Mol. Spectrosc. 55, 175 (1975). (1975). "G. D. Patterson (private communication). 3~.Behringer in Molecular Spectroscopy edited by R. F. Bar- "R. Z. Bachrach, Rev. Sci. Instrum. 43, 734 (1972). row, D. A. Long, and D. J. Miller (The Chemical Society, 29~.Ezekiel and R. Weiss, Phys. Rev. Lett. 20, 91 (1968); London, 1974), Vol. 2, pp. 100-172. and G. A. Capelle and H. P. Broida, J. Chem. Phys. 58, 4~.Behringer, J. Raman Spectrosc. 2, 279 (1974). 4212 (1973). 5~.V. Klein, Phys. Rev. B 8, 919 (1973). 31~ee,for example, K. B. McAfee and R. S. Hozack, J. Chem. 'see, for example, J. A. Coxon in Molecular Spectroscopy Phys. 64, 2491 (1976). (The Chemical Society, London, 1973), Vol. 1, pp. 177-228. 32~.Kiefer and H. J. Bernstein, J. Mol. Spectrosc. 43, 366 'A. C. Albrecht, J. Chem. Phys. 34, 1476 (1961). (1972). 'J. Tang and A. C. Albrecht in Raman Spectroscopy, edited by 33~.Tellinghuisen, J. Chem. Phys. 58, 2821 (1973). H. A. Szymanski (Plenum, New York, 1970), Vol. 2, pp. 34~.I. Steinfeld, R. N. Zare, L. Jones, and W. Klemperer, 33-68. J. Chem. Phys. 42, 25 (1965). 9~.P. Shorygin, Sov. Phys. -Usp. 16, 99 (1973). 35~.N. Zare, J. Chem. Phys. 40, 1934 (1964); R. Verma, 1°s. D. Silverstein, General Electric Technical Report J. Chem. Phys. 32, 738 (1960). #73CRD195, June 1973; S. D. Silverstein and R. L. St. 36~.W. Cooley, Math. Computation 15, 363 (1961). Peters, Chem. Phys. Lett. 23, 140 (1973); S. D. Silver- 37~.Herzberg, Molecular Spectra and Molecular Structure stein and R. L. St. Peters, Phys. Rev. A 9, 2720 (1974). (Van Nostrand-Reinhold, New York, 1950), p. 427. "M. Jacon and D. Van Labeke, Mol. Phys. 29, 1241 (1975). 3a~ualitativelysimilar calculations were made by M. Berjot, ''since the initial results of this calculation were reported p. M. Jacon, and L. Bernard [Opt. Commun. 4, 246 (1971)], F. Williams, D. L. Rousseau, and S. H. Dworetsky, Phys. but they only included contributions from v" =O and 1 in their Rev. Lett. 32, 196 (1974)1, several other calculations of calculations. lifetime responses using a variety of incident pulse shapes 39~.Placzek and E. Teller, Z. Phys. 81, 209 (1933). have been published and a prior report has been brought to 40~.L. Rousseau and P. F. Williams, Phys. Rev. Lett. 33, our attention. These include S. D. Silverstein, General 1368 (1974). Electric Technical Report #73CRD196, June 1973; J. M. 4i~.J. Oldman, R. K. Sander, and K. R. Wilson, J. Chem. Friedman and R. M. Hochstrasser, Chem. Phys. 6, 155 Phys. 54, 4127 (1971). (1974); J. 0. Berg, C. A. Langhoff, and G. W. Robinson, 42~.M. Schulman and R. Detrano, Phys. Rev. A 10, 1192 Chem. Phys. Lett. 29, 305 (1974); R. C. Hilborn, Chem. (1974).

J. Chem. Phys., Vol. 64, No. 9, 1 May 1976